Legendre, Hermite and UNIT 3 LEGENDRE, HERMITE AND Laguerre Polynomials LAGUERRE POLYNOMIALS Structure Page No. 3.1 Introduction 83 Objectives 3.2 Legendre Polynomials 84 Rodrigue’s Formula Generating Function Recurrence Relations Orthogonality Property Legendre Function of the Second Kind 3.3 Hermite and Laguerre Polynomials 100 3.4 Applications to Physical Situations 106 3.5 Summary 111 3.6 Solutions/Answers 112 3.1 INTRODUCTION In Unit 2, we discussed methods of finding series solutions of second order, linear, homogeneous differential equations with variable coefficients. In this unit and the unit to follow, we shall apply these methods to a few special second order equations occurring frequently in applied mathematics, engineering and physics. These equations are Legendre’s, Bessel’s, Hermite’s and Laguerre’s equations. The solution to these equations, that occur in applications, are referred to as special functions . They are called ‘special’ as they are different from the standard functions like sine, cosine, exponential, logarithmic etc. In this unit, we shall concentrate on Legendre, Hermite and Laguerre polynomials which are polynomial solutions to Legendre’s, Hermite’s and Laguerre’s differential equations. Legendre polynomials first arose in the problem of expressing the Newtonian potential of a conservative force field in an infinite series involving the distance variable of two points and their included central angle. Other similar problems dealing with either gravitational potential or electrostatic potentials and steady-state heat conduction problems in spherical solids, also lead to Legendre polynomials. Other polynomials which commonly occur in applications are the Hermite and Laguerre. They play an important role in quantum mechanics, and in probability theory. We have started the unit by obtaining the power series solutions of Legendre’s differential equation in Sec.3.2 and introduced Legendre polynomials and Legendre functions of both first and second kind. Various properties of Legendre polynomials are also discussed in this section. Polynomials solutions of Hermite’s and Laguerre’s equations and their properties are discussed in Sec.3.3. Applications of the Legendre and Hermite polynomials to physical situtations are discussed in Sec.3.4. Objectives After studying this unit you should be able to • obtain the power series solutions of Legendre’s differential equation; • derive Rodrigue’s formula, for Legendre polynomials; • obtain Legendre polynomials through generating function; • use recurrence relations for Legendre polynomials and its orthogonality property in various applications; • derive Rodrigue’s formula, generating function, recurrence relations and orthogonal property of Hermite and Laguerre polynomials and use them in various applications. 83 Ordinary Differential Equations 3.2 LEGENDRE POLYNOMIALS The differential equation of the form d 2 y dy 1( − x 2 ) − 2x + n(n + y)1 = 0 (1) dx 2 dx is called Legendre’s differential equation or simply Legendre’s equation , where ‘n’ is a real constant. We can also write Legendre’s equation in the form d dy 1( − )x 2 + n(n + y)1 = 0 (2) dx dx In Eqn.(1), if we divide by 1( − x 2 ) throughout then we can write the equation as d 2 y 2x dy 1 − + n(n + y)1 = ,0 x ≠ ± .1 (3) dx 2 1− x 2 dx 1− x 2 2x n(n + )1 Here − and are analytic functions for x = 0 . Also if we use 1− x 2 1− x 2 Binomial expressions, then both these converge for |x| <1 . Hence x = 0 is an ordinary point of Legendre’s Eqn.(1) and this suggests that Eqn.(1) has a power series solution about x = 0 . Assume the series solution ∞ k )x(y = ∑ c k x (4) k=0 Differentiting Eqn.(4) w.r. to x , we get ∞ k−1 y′ )x( = ∑ c k k x k=1 ∞ k−2 and, y ′′ )x( = ∑ c k k(k − )1 x k=2 Substituting for ),x(y y′ )x( and y′′ )x( in Eqn.(1), we get ∞ ∞ ∞ 2 k−2 k−1 k 1( − x )∑c k k(k − x)1 − 2x ∑c k k x + n(n + )1 ∑ c k x = 0 k=2 k= 1 k= 0 ∞ ∞ ∞ ∞ k−2 k k k i.e., ∑c k k(k − x)1 − ∑c k k(k − x)1 − 2 ∑c k kx + n(n + )1 ∑ c k x = 0 k=2 k= 2 k= 1 k= 0 ∞ ∞ ∞ k k+2 k+1 i.e., ∑c k+2 k( + k()2 + x)1 − ∑c k+2 k( + k()2 + x)1 − 2 ∑ c k+1 k( + x)1 k=0 k=0 k=0 ∞ k + n(n + )1 ∑ c k x = 0 (5) k=0 Equating the coefficients of various powers of x on both sides of Eqn.(5), we get n(n + )1 For x 0 : c )1.2( + n(n + c)1 = 0 .,e.i c = − c (6) 2 0 2 2 0 2 − n(n + )1 For x1 : c )2.3( − 2 c + n(n + c)1 = 0 .,e.i c = c (7) 3 1 1 3 3.2 1 k For x : c k+2 k( + k()2 + )1 − c k k(k − )1 − 2 c k k + n(n + c)1 k = 0 k(k − )1 + 2k − n(n + )1 .,e.i c = c , for k ≥ 0 k +2 k( + k()2 + )1 k k( − k()2 − )1 − n(n + )1 .,e.i c = c , for k ≥ 2 k k(k − )1 k −2 − n[ − k( − n[])2 + k( − ])1 = c , for k ≥ 2 (8) k k( − )1 k−2 84 From recurrence relation (8), alongwith Eqns.(6) and (7), we observe that coefficients Legendre, Hermite and Laguerre Polynomials c k for k even are multiples of c 0 while those for k odd are multiples of c1 . Thus, in general, we have − n( − 2k + n()2 + 2k − )1 c = c , for k ≥ 1 2k 2k 2( k − )1 2k−2 n( − 2k + n()2 − 2k + )4 K n( − ).2 n(n + n()1 + )3 L n( + 2k − )1 = (− )1 k c , 2( )!k 0 for k ≥1 (9) − n( − 2k + n()1 + 2 )k and, c = c , for k ≥ 1 2k+1 2( k + 2()1 )k 2k−1 n( − 2k + n()1 − 2k + )3 K n( − n()1 + n()2 + )4 K n( + 2 )k = (− )1 k c , 2( k + )!1 1 for k ≥ 1 (10) Substituting from relations (9) and (10) in relation (4), the power series solution of Eqn.(1) can be written as n(n + )1 2 n( + n()3 + n)1 n( − )2 4 L )x(y = c0 1 − x + x − )!n( )!4( n( + n()2 − )1 3 n( + n()4 + n()2 − n()1 − )3 5 L + c1 x − x + x − )!3( )!5( ∞ L K k n( − 2k + n()2 − 2k + )4 n( − ).2 n(n + n()1 + )3 n( + 2k − )1 2k = c 0 1 + ∑ (− )1 x k=1 2( )!k ∞ L K k n( − 2k + n()1 − 2k + )3 n( −1).( n + n()2 + )4 n( + 2 )k 2k+1 = c1 x + ∑ (− )1 x k=1 2( k + )!1 = c0 y1 )x( + c1 y 2 )x( (11) where, ∞ n( − 2k + n()2 − 2k + )4 K n( − ).2 n(n + n()1 + )3 K n( + 2k − )1 k 2k 12) y1 )x( = 1+ ∑ (− )1 x k=1 2( )!k and ∞ K K k n( − 2k + n()1 − 2k + )3 n( − ).1 n(n + n()2 + )4 n( + 2 )k 2k+1 (13) y2 )x( = x + ∑(− )1 x k=1 2( k + )!1 when n is not an integer both y1 )x( and y 2 )x( converge for |x| < 1. Since c 0 and c1 are arbitrary constants, and y1 and y 2 are linearly independent solutions of Eqn.(1), solution (11) can be considered as the general solution of Legendre’s Eqn.(1). The functions defined by Eqns.(12) and (13) are called Legendre Functions. In case, n is a non-negative integer, then y1 )x( reduce to a polynomial of degree n , if n even; and the same is true for y 2 )x( if n is odd. For example, for n even , we have For n = 0 : y1 )x( =1 2 For n = 2 : y1 )x( =1 − 3x 35 For n = 4 : y )x( =1 −10 x 2 + x 4 1 3 and so on. Thus y1 )x( reduces to a polynomial of even powers. For these values of y,n 2 )x( remains an infinite series. 85 Ordinary Differential In general, for n = 2m , where m is an integer, Equations m K K k 2( m − 2k + 2()2 m − 2k + )4 2m 2( m + 2()1 m + )3 2( m + 2k − )1 2k (14) y1 )x( = 1+ ∑(− )1 x k=1 2( )!k and for k = m + k,1 = m + ,2 K etc. the terms in the summation on the R.H.S. of Eqn.(14) are zero. When n is odd , y1 )x( remains an infinite power series and y 2 )x( reduces to polynomial of odd powers. For example, For n = 1 : y 2 )x( = ,x 5 For n = 3: y )x( = x − x 3 . 2 3 14 21 For n = 5 : y )x( = x − x 3 + x 5 . 2 3 5 In general, for n = 2m +1 , we may obtain ∞ K K k 2( m − 2k + 2()2 m − 2k + )4 2m.( 2m + 2()3 m + )5 2( m + 2k + )1 2k+1 . (15) y2 )x( = x + ∑(− )1 x k=1 2( k + )!1 These polynomials multiplied by suitable constants are called Legendre polynomials. The Legendre polynomials are denoted by Pn )x( where n denotes the order of the polynomial. Therefore, when n takes positive integral values, one of the linearly independent solutions of Eqn.(1) is a Legendre polynomial and the second solution is an infinite series.